Introduction to the Fast Multipole Method: Topics in Computational Biophysics, Theory, and Implementation, 1st Edition (Hardback) book cover

Introduction to the Fast Multipole Method

Topics in Computational Biophysics, Theory, and Implementation, 1st Edition

By Victor Anisimov, James J.P. Stewart

CRC Press

448 pages | 73 B/W Illus.

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Introduction to the Fast Multipole Method introduces the reader to the theory and computer implementation of the Fast Multipole Method. It covers the topics of Laplace’s equation, spherical harmonics, angular momentum, the Wigner matrix, the addition theorem for solid harmonics, and lattice sums for periodic boundary conditions, along with providing a complete, self-contained explanation of the math of the method, so that anyone having an undergraduate grasp of calculus should be able to follow the material presented. The authors derive the Fast Multipole Method from first principles and systematically construct the theory connecting all the parts.

Key Features

  • Introduces each topic from first principles
  • Derives every equation presented, and explains each step in its derivation
  • Builds the necessary theory in order to understand, develop, and use the method
  • Describes the conversion from theory to computer implementation
  • Guides through code optimization and parallelization

Table of Contents

1. Legendre Polynomials

1.1 Potential of a Point Charge Located on the z-Axis

1.2 Laplace’s Equation

1.3 Solution of Laplace’s Equation in Cartesian Coordinates

1.4 Laplace’s Equation in Spherical-Polar Coordinates

1.5 Orthogonality and Normalization of Legendre Polynomials

1.6 Expansion of an Arbitrary Function in Legendre Series

1.7 Recurrence Relations for Legendre Polynomials

1.8 Analytic Expressions for First Few Legendre Polynomials

1.9 Symmetry Properties of Legendre Polynomials

2. Associated Legendre Functions

2.1 Generalized Legendre Equation

2.2 Associated Legendre Functions

2.3 Orthogonality and Normalization of Associated Legendre Functions

2.4 Recurrence Relations for Associated Legendre Functions

2.5 Derivatives of Associated Legendre Functions

2.6 Analytic Expression for First Few Associated Legendre Functions

2.7 Symmetry Properties of Associated Legendre Functions

3. Spherical Harmonics

3.1 Spherical Harmonics Functions

3.2 Orthogonality and Normalization of Spherical Harmonics

3.3 Symmetry Properties of Spherical Harmonics

3.4 Recurrence Relations for Spherical Harmonics

3.5 Analytic Expression for the First Few Spherical Harmonics

3.6 Nodal Properties of Spherical Harmonics

4. Angular Momentum

4.1 Rotation Matrices

4.2 Unitary Matrices

4.3 Rotation Operator

4.4 Commutative Properties of the Angular Momentum

4.5 Eigenvalues of the Angular Momentum

4.6 Angular Momentum Operator in Spherical Polar Coordinates

4.7 Eigenvectors of the Angular Momentum Operator

4.8 Characteristic Vectors of the Rotation Operator

4.9 Rotation of Eigenfunctions of Angular Momentum

5. Wigner Matrix

5.1 The Euler Angles

5.2 Wigner Matrix for j = 1

5.3 Wigner Matrix for j = 1/2

5.4 General Form of the Wigner Matrix Elements

5.5 Addition Theorem for Spherical Harmonics

6. Clebsch–Gordan Coefficients

6.1 Addition of Angular Momenta

6.2 Evaluation of Clebsch–Gordan Coefficients

6.3 Addition of Angular Momentum and Spin

6.4 Rotation of the Coupled Eigenstates of Angular Momentum

7. Recurrence Relations for Wigner Matrix

7.1 Recurrence Relations with Increment in Index m

7.2 Recurrence Relations with Increment in Index k

8. Solid Harmonics

8.1 Regular and Irregular Solid Harmonics

8.2 Regular Multipole Moments

8.3 Irregular Multipole Moments

8.4 Computation of Electrostatic Energy via Multipole Moments

8.5 Recurrence Relations for Regular Solid Harmonics

8.6 Recurrence Relations for Irregular Solid Harmonics

8.7 Generating Functions for Solid Harmonics

8.8 Addition Theorem for Regular Solid Harmonics

8.9 Addition Theorem for Irregular Solid Harmonics

8.10 Transformation of the Origin of Irregular Harmonics

8.11 Vector Diagram Approach to Multipole Translations

9. Electrostatic Force

9.1 Gradient of Electrostatic Potential

9.2 Differentiation of Multipole Expansion

9.3 Differentiation of Regular Solid Harmonics in Spherical Polar Coordinates

9.4 Differentiation of Spherical Polar Coordinates

9.5 Differentiation of Regular Solid Harmonics in Cartesian Coordinates

9.6 FMM Force in Cartesian Coordinates

10. Scaling of Solid Harmonics

10.1 Optimization of Expansion of Inverse Distance Function

10.2 Scaling of Associated Legendre Functions

10.3 Recurrence Relations for Scaled Regular Solid Harmonics

10.4 Recurrence Relations for Scaled Irregular Solid Harmonics

10.5 First Few Terms of Scaled Solid Harmonics

10.6 Design of Computer Code for Computation of Solid Harmonics

10.7 Program Code for Computation of Multipole Expansions

10.8 Computation of Electrostatic Force Using Scaled Solid Harmonics

10.9 Program Code for Computation of Force

11. Scaling of Multipole Translations

11.1 Scaling of Multipole Translation Operations

11.2 Program Code for M2M Translation

11.3 Program Code for M2L Translation

11.4 Program Code for L2L Translation

12. Fast Multipole Method

12.1 Near and Far Fields: Prerequisites for the Use of the Fast Multipole Method

12.2 Series Convergence and Truncation of Multipole Expansion

12.3 Hierarchical Division of Boxes in the Fast Multipole Method

12.4 Far Field

12.5 NF and FF Pair Counts

12.6 FMM Algorithm

12.7 Accuracy Assessment of Multipole Operations

13. Multipole Translations along the z-Axis

13.1 M2M Translation along the z-Axis

13.2 L2L Translation along the z-Axis

13.3 M2L Translation along the z-Axis

14. Rotation of Coordinate System

14.1 Rotation of Coordinate System to Align the z-axis with the Axis of Translation

14.2 Rotation Matrix

14.3 Computation of Scaled Wigner Matrix Elements with Increment in Index m

14.4 Computation of Scaled Wigner Matrix Elements with Increment in Index k

14.5 Program Code for Computation of Scaled Wigner Matrix Elements Based on the k-set

14.6 Program Code for Computation of Scaled Wigner Matrix Elements Based on the m-set

15. Rotation-Based Multipole Translations

15.1 Assembly of Rotation Matrix

15.2 Rotation-Based M2M Operation

15.3 Rotation-Based M2L Operation

15.4 Rotation-Based L2L Operation

16. Periodic Boundary Condition

16.1 Principles of Periodic Boundary Condition

16.2 Lattice Sum for Energy in Periodic FMM

16.3 Multipole Moments of the Central Super-Cell

16.4 Far-Field Contribution to the Lattice Sum for Energy

16.5 Contribution of the Near-Field Zone into the Central Unit Cell

16.6 Derivative of Electrostatic Energy on Particles in the Central Unit Cell

16.7 Stress Tensor

16.8 Analytic Expression for Stress Tensor

16.9 Lattice Sum for Stress Tensor

About the Authors

Victor Anisimov is an Application Performance Engineer at the Argonne National Laboratory

James J. P. Stewart is the author of the MOPAC program developed by Stewart Computational Chemistry, LLC

Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Life Sciences / Biology / General
SCIENCE / Chemistry / General
SCIENCE / Chemistry / Physical & Theoretical
SCIENCE / Physics